The exponent of $p$ in $x^{p-1}-1$: Yamada's constant 283 lowered to 69

Yamada (J. Number Theory 130, 2010) deduced from the method of interpolation determinants of Bugeaud and Laurent the bound $v_p(x^{p-1}-1)\le 283\,(p-1)\log y\,\log(xy)/(\log p)^2+4$, for multiplicatively independent integers $x,y$ coprime to $p$. We replace 283 by 69. The new ingredient is an elementary bound, by rearrangement, on the spread of the exponents that occur in the Liouville inequality: it at least halves the height term of Yamada's criterion, involves no congruence class, and allows the parameters to be re-optimised. The proof is complete up to the lemmas of Bugeaud–Laurent (zero lemma, analytic upper bound) and of Laurent–Mignotte–Nesterenko (bound for the polynomial), which are used as they stand. A self-contained verifier checks the numerical inequalities.